Search for Periodic Orbits in Delay Differential Equations

2014 ◽  
Vol 24 (06) ◽  
pp. 1450084 ◽  
Author(s):  
Romina Cobiaga ◽  
Walter Reartes
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Periodic Orbits ◽  
Heuristic Search ◽  
Delay Differential Equations ◽  
Anharmonic Oscillator ◽  
Analysis Method ◽  
Van Der Pol ◽  
Homotopy Analysis ◽  
Delay Differential

In a previous paper, we developed a new way to apply the Homotopy Analysis Method (HAM) in the search for periodic orbits in dynamical systems modeled by ordinary differential equations. This method differs from the original in the heuristic search of the frequencies of the cycles. In this paper, we show that the method can be extended to the search for periodic orbits in delay differential equations. Herein, this methodology is applied twice, firstly in an equation of van der Pol type and secondly in an anharmonic oscillator, both systems with a delayed feedback.

scholarly journals A Natural Homotopy Analysis Method for Nonlinear Delay Differential Equations

2019 ◽  
Vol 1 (2) ◽  
pp. 86-90
Author(s):  
Aminu Barde
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Delay Differential Equations ◽  
Analysis Method ◽  
Linear Delay ◽  
Homotopy Analysis ◽  
Nonlinear Delay Differential Equations ◽  
Functional Differential ◽  
Delay Differential

Delay differential equation (DDEs) is a type of functional differential equation arising in numerous applications from different areas of studies, for example biology, engineering population dynamics, medicine, physics, control theory, and many others. However, determining the solution of delay differential equations has become a difficult task more especially the nonlinear type. Therefore, this work proposes a new analytical method for solving non-linear delay differential equations. The new method is combination of Natural transform and Homotopy analysis method. The approach gives solutions inform of rapid convergence series where the nonlinear terms are simply computed using He's polynomial. Some examples are given, and the results obtained indicate that the approach is efficient in solving different form of nonlinear DDEs which reduces the computational sizes and avoid round-off of errors.

scholarly journals THE HOMOTOPY ANALYSIS METHOD IN BIFURCATION ANALYSIS OF DELAY DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 22 (08) ◽  
pp. 1230024 ◽  
Author(s):  
ANDREA BEL ◽  
WALTER REARTES
Keyword(s):  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Homotopy Analysis Method ◽  
Delay Differential Equations ◽  
Analysis Method ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Homotopy Analysis ◽  
Delay Differential ◽  
Bifurcation And Stability

In this paper we apply the homotopy analysis method (HAM) to study the van der Pol equation with a linear delayed feedback. The paper focuses on the calculation of periodic solutions and associated bifurcations, Hopf, double Hopf, Neimark–Sacker, etc. In particular, we discuss the behavior of the systems in the neighborhoods of double Hopf points. We demonstrate the applicability of HAM to the analysis of bifurcation and stability.

THREE TYPES OF SIMPLE DDE'S DESCRIBING TUMOR GROWTH

2007 ◽  
Vol 15 (04) ◽  
pp. 453-471 ◽  
Author(s):  
MAREK BODNAR ◽  
URSZULA FORYŚ
Keyword(s):  
Experimental Data ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Tumor Growth ◽  
Delay Differential Equations ◽  
Ehrlich Ascites Tumor ◽  
Ascites Tumor ◽  
Ehrlich Ascites ◽  
Delay Differential

In this paper, we compare three types of dynamical systems used to describe tumor growth. These systems are defined as solutions to three delay differential equations: the logistic, the Gompertz and the Greenspan types. We present analysis of these systems and compare with experimental data for Ehrlich Ascites tumor in mice.

scholarly journals The Oscillation of a Class of the Fractional-Order Delay Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qianli Lu ◽  
Feng Cen
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Analysis Method ◽  
Constant Coefficients ◽  
Delay Differential ◽  
Necessary And Sufficient

Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this,α-exponential function which is a kind of functions that play the same role of the classical exponential functions of fractional-order derivatives is used.

Control of Chaos in Networks with Delay: A Model for Synchronization of Cortical Tissue

1994 ◽  
Vol 6 (6) ◽  
pp. 1141-1154 ◽  
Author(s):  
C. Lourenço ◽  
A. Babloyantz
Keyword(s):  
Differential Equations ◽  
Periodic Orbits ◽  
Delay Differential Equations ◽  
Chaotic Dynamics ◽  
Cortical Activity ◽  
Inhibitory Neurons ◽  
Unstable Periodic Orbits ◽  
Cortical Tissue ◽  
Control Of Chaos ◽  
Delay Differential

The unstable periodic orbits of chaotic dynamics in systems described by delay differential equations are considered. An orbit is stabilized successfully, using a method proposed by Pyragas. The system under investigation is a network of excitatory and inhibitory neurons of moderate size, describing cortical activity. The relevance of the results for synchronized cortical activity is discussed.

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